How many days off are allowed for a live-in caregiver?

No. She absolutely cannot "do that". You are not a bond servant whose indenture she purchased.
My question is why did you not discuss what your schedule for time off would look like before you took the position and moved in? Even if you signed a contract agreeing to the two weeks off a year, that's vacation time. Vacation time is different than the days off a person gets during the week. If you plan on making your living as a private caregiver, you've got a lot to learn my friend. Like rule Number 1: Never take a caregiving assignment where the client's home is your actual residence because you will always get taken advantage of. Also, the employer can dismiss you straight away regardless of whether or not you have somewhere else to go. People can fire caregivers and not have to do an eviction process many states. You are at their mercy. Don't leave yourself vulnerable like that.
Rule Number 2: Payday is always on the same day. Wages paid in full. NO EXCEPTIONS. Never give an employer an inch when it comes to your pay and when you get it. People think it's okay to stiff workers in the care profession on their pay and you can never allow them to. Anyone I ever worked for knew they could count on me to do the best care for the time I'm being paid to do it. That I would get to the client's home no matter what. They also knew that if my money isn't right where it's supposed to be right when it's supposed to be there, I will walk away from their LO regardless of their needs and we are done.
Keep these rules in mind when you take work.
Great that you live in Connecticut. So do I. This state has a terrific Labor Department.
It is illegal for you to be working seven days a week with no time off. Being allowed to go to bed at night is not considered time off.
As a live-in you are entitled to at least one day off a week. A day off means 24 hours. You can leave your job and sleep elsewhere. That is a day off.
Tell your employer that either you get one or two FULL days off a week, or work out a time off plan like you work 'X' number of days on then get 'X' number of days off.
Then you give your employer a choice. Either the time off plan becomes effective immediately as in this very week, or you will abandon the client and head straight for the Connecticut Department of Labor and report this illegal situation to them.
I'm pretty sure our employer will get another caregiver to work your time off or they will cover it themselves.

Did an attorney draw up the contact? It sounds like it isn't legal. Be prepared to lose your job if you insist on the employer make everything in line with federal wage and hour standards.

Your employer is wrong. Talk to the Labor board. This has been discussed before on this forum so this is where I am getting my info from but again, check with your Labor board.

You cannot be worked more than 40 hrs. You maybe entitled to overtime if you do.

You need to be given time off, away from the home. There are set hours. Your boss needs to do the caregiving while ur away or hire someone else.

Room and board are a perk and cannot be deducted from ur pay.

Be ready to lose your job if you find I am right and you confront your boss with what the Labor laws are in your State.

When you are "off-duty", are there other family in the home to take over care?

At those "off-duty" hours, are you allowed to leave the premises?

If this misunderstanding (boss breaking the law) ends in a dispute with her firing you there are things you can do. You say taxes are being withdrawn
(taxes withheld). Check to see if the taxes are being paid quarterly to the IRS on your behalf. Call Social Security to see if payments to your account have been posted. You may need an attorney to sort this out.

In the meantime, do not just walk away, quit, or abandon your client who needs care. If you must leave for your health, call APS to investigate with a home visit. It is clear that your boss has misinterpreted the rules/laws.

And what a shame for your client, because good caregivers are hard to find!

Adding this:
I do not know of a client/patient needing daily caregiving that would not have any needs in the middle of the night. Who takes care of your client overnight?

If you are not allowed to leave because your client can't be alone then you ARE working 24 hours; just as firefighters and EMS get paid when they are on call but there isn't an emergency or babysitters get paid after the children are in bed you may not be actively giving care but you are on duty. Contracts have no legal weight when they break the law, and no labour code allows for 24/7 employment without respite.

Call the Department of Labor. Do you work 24 hours? That is slavery! Are you paid overtime?

Did you sign a contract? If so, what does it say about time off?

My friend's MIL has a live-in caregiver, paid under the table. The caregiver takes off every other Friday evening and doesn't return until the following morning Monday morning.