If an employer provides a check to be mobile deposited and subsequently the check is insufficiently funded, and creating an overdraft of my account. The employer "claims" in order to resolve the overdraft, I need to provide my personal account detail access: username, password, routing, and account numbers. Is this legal?
This is a scam. What you should do is cut off contact with them immediately.
No employer ever needs your bank username and password. Period.
Even the bank employees will not ask you for your password.
(Routing and account numbers are a different matter; that can be legitimate information for direct deposit, and you give people those every time you write a check.)
If you have already given them this information, you need to CONTACT YOUR BANK IMMEDIATELY. They may have a fraud hotline that operates 24/7; if so, call them now.
Giving your password to them violates your agreement with your bank.
I agree with the others. This has 100% of the markings of an overpayment scam - that's where they send you more money than you're entitled to, and then ask you to send some of it back. They ask you to send the money back via a non-reversible means such as Western Union, Zelle, gift card numbers, or Bitcoin. And after that, their payment to you bounces or reverses - imagine that!
If you are overdrawn because you sent money to this employer or someone they told you to send money to, then this is the scam that happened to you.
Sometimes they use a forged paper check which is rigged to pinball all around Africa before it finally bounces, by which time the bank has released the "hold" on the money and you think "oh, it has cleared, right?" Sometimes they use money from some other victim's bank account whose username, password and details they have (gee, how'd they get that?) ... and later the victim discovers the withdrawal and reverses it and you're out the money.
The entire point of the scam is the money you sent back. Everything else is was. window dressing on an elaborate scam aimed at getting you to believe the deal was real, to get you to send the money back.
They will try to string you along until it's obvious there's no more money to get out of you, and then they will ghost you. Asking for your bank details is probably so they can look in your account and see what money you have in accounts, credit card limits, IRAs and the like. Certainly they will sign you up for Zelle and try to transfer out any money they find.
And to get it across five x five, there is no job here. This is not an employer, it's a scammer who took you. No legitimate employer will send a new employee a bunch of money "blind" and ask you to send some of it back. That does not happen. A real employer who did that would be a fool.
Also I'm sure you've applied and been accepted for real jobs in real, bricks and mortar places that actually exist. So you know the list of documents that they always ask for. They are obliged to check your "right to work" (citizenship or visa), legal age, etc. Never offer this. Wait for them to ask, because if they don't ask, that is a "red flag" for some sort of scam or ripoff.
Contact the police
Other answers have already covered protecting yourself from fraud.
One thing that hasn't been covered though is that these people are criminals. In many countries, simply what they've done with you is open-and-shut illegal. But it's highly unlikely that you're the first one. Once they're arrested, it'll likely be fairly easy for the police to trace all the other people they've defrauded.
So call the police, and get them put away.
Why do they need that information to resolve the overdraft? All they have to do to resolve the overdraft is give you the money (e.g. with a check that doesn't bounce).
You never, ever need to give someone your username and password to have a financial transaction with them. There are numerous mechanisms for payment, none of which require either party to have the other's username and password. I've issued thousands of payments over the years and never had to give anyone that.
That being said, this is definitely a scam. A legitimate employer wouldn't ask you for that information.