In Latin the suffix -que can be used to mean "and". For example:
Fames sitisque (Hunger and thirst)
Are there any modern Romance languages that use the suffix -que or something similar to it?
This claims that Romanian şi (and) derives from "-que", but Wiktionary derives it from "sic" (thus). (Being Wiktionary, of course without any sources.) To me, going from thus to and seems like a big leap, but I don't know anything for sure about that. Edit: Apparently not that giant a leap, see comments.
As far as actual enclitics go, I'm not aware of any Romance language that uses one, but I only know about the 'big' ones (Italian, French, Castilian, Portuguese, Romanian).
Two French/Italian examples :
o French "quiconque" (anyone) < Latin quicumque (anyone)
quicumque = qui + cum + -que
Italian : chiunque
o French "chacun" (each one) < Vulgar Latin "cascunu" < Latin "quisque unus" (every one), maybe mixed with Ancient Greek "unum kata unum" (one by one).
quisque = quis + -que
If 'something similar' to Latin -que is good enough for you, present-day Spanish conmigo contains an old preposition or prepositional clitic suffix -cum hidden under the current from -go. [Our conmigo actually derives from cum mecum, where the -cum part of mecum was originally an adposition/a syntactic affix meaning 'with' arguably similar to the -que clitic meaning 'and'. That construction was, of course, redundant, a remnant of the stage at which late spoken Latin was becoming a head-complement language, what Romance languages soon were to develop into, but still contained important residues, mainly in VPs and PPs, of the earlier complement-head phase]. Less obviously, present-day Spanish quisqui, as in todo quisqui (= every body), cada quisqui (= each person), is a reflex of Latin quis-que (originally perhaps = somebody else).