In order to address the broader question in your title,
Should I be Wary of First-Time Home Ownership Programs?
It's important to point out that there are a huge variety of (legitimate) housing programs in the US which aren't well known, and which sometimes offer (completely legitimate) subsidies or features that are surprisingly attractive. Often these programs are sponsored through government programs or grants, which give lenders some leeway in terms of writing loans that otherwise wouldn't be possible. Sometimes these programs are run by state or local government programs, which explains why they may be very small in scope, variable in how they work, and relatively unknown. Arguably, FHA, USDA, and VA loans are the most well known housing programs in the US, but there are often other programs in a given market. To me, it sounds like the loan you're being referred to might be one of these programs.
The market for loans is driven on decisions about risk. A lender's loan offerings and their underwriting rules are essentially their translation of risk into dollars. If a lender chooses not to offer a certain product to a certain customer, it's basically because doing so would be too risky. Hence, a program that works by offsetting risk with cash in some way can allow lenders to offer loans that they otherwise wouldn't make.
That's where special programs come in to play. Typically, these programs are designed to incent certain segments of the population into buying homes. This is implemented through subsidizing the lender for writing those loans. The money that comes from these programs basically allows the lender to offset risk that might normally be associated with a particular loan for a particular individual. There are a dizzying array of ways these programs are actually carried out (sometimes the bank doesn't hold the loan - the government does. Sometimes the grant program directly contributes cash to the loan as a pseudo-downpayment. Sometimes the grant program directly contributes cash to the lender which effectively acts as a pad on the interest rate. Sometimes the lender gets access to a pool of money as a reserve against losses. And so on).
As an example, my credit union recently had a program where they received a grant to encourage a segment of the population to buy homes, who otherwise wouldn't have been able to. The government was interested in increasing the number of homeowners in a certain segment of the population, but obviously didn't want to create loans that were extra risky, so the grant was used to help encourage good habits (saving) and also to offset risk. Members needed to pass a typical loan application process, and then they needed to contribute a fixed deposit amount to a special savings account every month for a year, to show that they were earnest about saving towards the home, and were able to budget money each month for home ownership. If they did that, the grant program matched their total deposit at the end of the year as a way of inflating their down payment. The loans were also written at a lower-than-typical interest rate and the credit union received an interest subsidy that offset the discount. Many of the people who participated in this program were able to walk into a new house with significant equity and a reasonable interest rate on a loan they could afford, whereas they wouldn't have been able to get an affordable loan (or any at all) without the program's help. It's a win-win-win: the customer gets help without being put in a risky situation, the credit union gets a new customer, and the local government gets a happy and stable homeowner paying taxes.
Eligibility requirements for these programs are usually in place to make sure they're used for the "right" people. This is why they will often have income caps, or only be available to customers in certain zip codes, or other restrictions.
Of course, all of that just provides a backdrop against which you can understand how programs can exist. None of us know if the loan you're being offered is legitimate or not. Which is why you're asking the obvious question,
How will I know if taking this loan is in my best interest?
Effectively, the answer to that is the same regardless of whether or not the loan is in a special program or not: Make sure you understand your own financial picture independently of what the bank tells you. Make sure you know what you can afford, what risk you are willing to assume, and what your options are if you need to change things in the future (i.e. unexpectedly sell the home).
The last point is important. Don't just make a decision based on if you can afford a monthly payment. Make sure you understand the "life" of the loan, and you have a legitimate exit strategy if life throws you a curve-ball. Ignoring this part of the decision is what usually gets people into trouble. People take out a loan that they can afford right now but in 2 years they lose their job, get divorced, or their spouse dies. Now, they either can't afford the payment, or they need to sell the home but find they can't easily do that because of how much they still owe on the loan.
Loans with no downpayment are risky in that sense because, with no equity in the house, it's likely you'll be upside-down if something bad happens within the first several years of the loan, especially if the housing market drops or stays stagnant during that period.
Also, as a minor footnote, some "special" loans are non conforming, which means they don't fit typical underwriting guidelines. This, unfortunately, gives lenders an opening to do damaging things with the terms of the loan, like adding in an early repayment penalty (which isn't allowed in a typical conforming loan). So, make sure you read the fine print and understand if there are any obstacles to an early exit from the loan.
So, step one is making sure you can afford the loan, and making sure you understand your options in the future, especially in terms of getting out of the loan quickly if you have to.
That's really the most important step to take. Secondly, though, it can be helpful to understand the "special" loan you're being offered. When you meet with the lender, ask why they're offering these special loans, and how they're able to offer you something outside of the "norm" in terms of mortgages. A lender who is participating in a legitimate housing program will be happy to explain that to a customer (these programs are usually very beneficial for a lender, so they're happy to spread the word). Or, you may get a shifty, redirected answer (which is obviously a red flag). Some less-trustworthy lenders even advertise their own (arguably predatory) loans in a way that makes them sound like a legitimate program, but they're not. We've probably all seen the clickbait adds in our web browsers with text that says something like, Obama wants everyone making under $100k to refinance now! That's a good example of a scammy copycat!
If it turns out that the loan is just a scam of some sort, and not a legitimate sponsored housing program, well then - at least hopefully you've learned something by reading this answer and asking questions. And maybe there is another lender in your area that does offer a legitimate loan you can take advantage of. The moral of the story is, don't assume that FHA loans are the only special programs out there.
Finally, at the end of the day, if there's anything you're unsure of, don't hesitate to review the paperwork or the loan terms with an advisor you trust. In some states in the US, it's typical to have a real estate attorney involved in a home purchase anyways, so if you have one, you can ask them to review the paperwork. If they're active in your area and aware of this lender, they may be able to explain the program and what it means from a more independent perspective than the lender can. Paying a lawyer to look at a contract may sound like a waste of money when you're trying to save up for a home, but buying a home is a big enough decision that it makes sense to be extra cautious before you commit to anything.