Can I itemize deductions one year, use 1040ez the following year and still retain carryover losses from the first year for use in the third year

What type of carryover losses?

carryover losses from stock trades

No, you can not do that. You must carry the stock loss to the very next tax year. Which does, in deed, require you to use form 1040.

But, Stock losses are totally unrelated to itemized deductions.  Itemized deductions go on Schedule A. Stock gains and losses, including carry over, go on Schedule D

I think OP added "in the third year" after I answered. Thus, I thought he was asking  if there was a way to show it on Form 1040EZ. It makes more sense now.

I think you need to file Form 1040 in order to use it the next year.