Absolutely yes, sign languages have rhymes and poetry and rhythm. All are based on modifying the prose form (normal, non-special daily use form) of the language to create heightened sensation - through control of pace, repetition of handshape or movement or facial features or accentuation of certain signs.
I'm not keen on the webpage @user6726 kindly linked to. I don't like signed poems based on spoken language poems, which often seem unnatural to me or to native signers (maybe because they are inevitably partial translations and not grounded in a visual heritage). For an alternate look at a more native signer style of verse, see this children's cowboy duel in visual vernacular. There's no subtitles but the general context is clear.
FYI, my deaf kids used to love watching this. Note the tightly controlled pace and rhythm, the strategic repetition of handshapes, the speedup and slowdown of signing, the way that several phases (aka lines) are brought back to the same situation / location, and many other elements. Searching Youtube for 'Deaf Cowboy' will bring up other videos in the same style.
Another accessible poem is 'Tree' by Paul Scott. It's easy to understand without subtitles even though it is in BSL (British Sign Language) not ASL (American Sign Language). You can see the use of poetic repetition here, also the poetic use of magnification to suddenly change focus from something big (the tree) to something small (the cat). Another poetic element: some of the signs used here are not signs normally used in every day speech - because they are awkward or would look contrived in everyday discourse. However, in poetry they fit.
Hope that helps?