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They can require you to take your lunch in the first example, if it is your regular lunch time. They can mandate that specific period for a lunch.
They can generally tell you not to work for any period of time that they wish, so they can require you to take a two or three hour lunch if they want.
I'm assuming that the point here, for them, is that if you are doing something medically related, then they want that PTO to be used. They don't want it horded, which is not an illegal motivation.
This is legal, because PTO is simply not regulated by state law in that fashion. If the employer has this as a policy for their PTO, that's not going to be illegal.
Now, you don't have to tell them that you are going to a medical appointment, so if you schedule one during your regular lunch break, you really should be able to do it without having them charge it against your PTO.
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