Depends on what your abilities are. Many homeowners DIY a faucet replacement.

It's probably the ceramic cartridge.

Most ceramic cartridges are lubricated wit a non-soluble lubricant that allows the two ceramic discs in the cartridge to move across each other smoothly.

But, even though the lubricant is not soluble, it wears away eventually from the scouring action of the water (and minerals in the water), making the cartridge hard to operate and sometimes seizing up altogether.

The squeaking you hear are the two discs rubbing together lubricant-less.

Some newer super cartridges such as those used in Delta, Brizo, and In2aqua faucets don't need lubricant, so they last a whole lot longer.

Replacing the cartridge should solve the problem. It's easy to do if you are at all handy and requires household tools you are likely to already own. Instructions usually come with the new cartridge and quite often there is a video or two on Youtube showing the process step by step.

If you are the original owner of the faucet, the replacement cartridge is free under Rohl's lifetime cartridge warranty.

Thanks very much for your informative reply. My husband will be attempting the repair and I just wanted to kind of have a heads up on which direction this might go. He is handy but has been procrastinating on tackling this...as we get older it seems to be a "thing." I keep asking him to take a look at it...he keeps not doing so. Ha! Anyway, we are the original owners so the info about the warranty is most helpful. Thanks again for your reply. I appreciate it very much.

Rohl sent me replacement ceramic cartridges for faucets that were 15 years old, no questions asked. But you will need the model name/number. (We still had the receipt in our bathroom remodel file).