I work a lot of overtime for my company. In my most recent pay period, I worked 8.5 hours of over time. When I got my pay stub, they did not give me my overtime pay, but gave me holiday pay for Memorial day. When I pointed out to them that I didn't get my overtime pay they said:
"Because it was a holiday week, and holiday hours cannot count towards OT pay, your pay it is correct. Please see the attached email with the guidelines for holiday pay. And note that #9 indicates that holiday pay is not to be considered hours worked in the computation of Overtime. And our overtime is calculated on hours worked over 40 in one week, not hours over 8 in a given day."

Is this legal? And if so, can I just not take holiday days such as Memorial day in order to avoid this happening again?

Holiday pay need not be counted as hours worked for purposes of determining overtime. If you actually worked over 40 hours in the work week, however, you must be paid overtime pay for the excess hours. They just are allowed to ignore the paid holiday hours; they aren't hours worked.

I'm not sure what giving up a paid holiday would buy you, however.

Is this legal? And if so, can I just not take holiday days such as Memorial day in order to avoid this happening again?

Legal, if you did not work more than 40 hours in the work week.

I do not understand the second question. NOT take the holiday off? And that will do what for you? You took a day off and got paid for NOT working and you are complaining?

Your employer has no legal obligation to allow you to work on a holiday. Your employer has no legal obligation to allow you to work any time they don't want you to work. It is their right to limit the hours you work so that you don't go into overtime; it is not your right to work whatever hours you want.

Overtime is not calculated on the basis of how many hours you were paid for. It is paid on the basis of how many hours you actually WORKED. If you did not work on the holiday, you are not entitled by law to have those hours included in overtime calculations. Nor does your state require overtime to be calculated on over 8 hours in a day, only over 40 hours in a week.

So unless you WORKED over 40 hours in a week (not, were paid for, WORKED) you were paid correctly.