Do USA iPhones work in Hungary?

"Work" -- can depend on what do you mean by work.

First, open https://support.apple.com/kb/sp685?locale=en_US for the iPhone 5s specifications.

Can you make phone calls? There are only four bands and you can see in the specs the phones cover all four: GSM/EDGE (850, 900, 1800, 1900 MHz) so you will have phone calls and very slow data everywhere in the world.

Speedier data is covered by the section UMTS/HSPA+/DC-HSDPA (850, 900, 1900, 2100 MHz). Now hop over to Wikipedia https://en.wikipedia.org/wiki/List_of_UMTS_networks#Europe to find that 900 / 2100 networks are used in Hungary and Europe in general. Compare to the list just given -- yay we have 3G support.

Even speedier data is covered by the section LTE (Bands 1, 2, 3, 4, 5, 8, 13, 17, 19, 20, 25). LTE is the jungle. You want https://en.wikipedia.org/wiki/List_of_LTE_networks_in_Europe the major providers use 3, 7, 20 (and a small one uses 31). This means LTE is a definite maybe: if you are in an area covered by only LTE Band 7 you won't have LTE otherwise you will. This is anyone's guess, providers rarely if ever provide per band coverage maps. A crowdsourced attempt can be found at http://gyebro.com/LTE/ which shows B7 is extremely rare and likely have coverage from B3 too.

All mentioned models work in Hungary (and anywhere in Europe).

Techincally, that should be clear anyway, but it is also personally verified this year.

Note that you need a data plan, and you are looking at roaming cost. The phones techincally work, and can use any of the local providers, but not for free of course.

All the phones you mention will work globally; however they may not be able to take advantage of the faster LTE network (if available in your area).