If you've got the money to pay the bill today, do it. They are giving you a 25% discount if you do. You won't find an investment that will beat that.
Let's look at the details of your scheme. Instead of paying $1696 today, you decide that you will pay $2261 over 60 months, or $37.68 per month. You also decide to invest $1696 today, and expect to get 6% return each year.
Your investment gets you $102 each year, but you have to pay taxes on that. If you are in the 25% tax bracket, you only keep $76 (ignoring state taxes). In addition, the loan is costing you $452 in payments each year.
At the end of the 5 years, you will have paid $2261 to the hospital, and your $1696 investment will be worth about $2123 after taxes.
Instead, let's say that you paid the hospital $1696 today, and invested the $37.68 per month. At the end of 5 years, assuming the same 6% growth and 25% tax bracket, your investment will be worth $2552.
In order for you to come out ahead by investing today and paying off the hospital over time, you would need to get at least a 17% growth on your investment. If you are ignoring taxes, then the number you need to hit is at least 13%.
Conclusion: You will come out ahead by paying the hospital today, and investing the monthly payment plan that you avoided.
(Note: Bankrate has a very handy investment calculator that makes it easy to calculate returns on a monthly investment.)
Now, let's look at the ethics of the situation. Assume that you were able somehow to find an investment with a guaranteed return high enough to come out ahead with your plan. Should you do it?
The hospital has provided you a service, and you owe the money. As a public service to people that cannot pay the bill, they allow people to pay off the bill over time at no interest. However, you are not one of these people. You have the money to pay. It is not ethical, in my opinion, to use the hospital's money to invest and try to profit.