Mmm, as far as I can tell none of the mentioned Russian verbs are etymologically related to Ukrainian чавити (NB cognate words are not translations, they are etymologically related words, possibly having a different meaning). Perhaps, just perhaps, давить is related but 1) it is not mentioned in the sources I consulted (e.g. викисловарь) and 2) the shift д->ч seems difficult to explain.Cognate words in Russian of the Ukrainian verb чавити: [...]
You have to distinguish šťaviť sa and staviť sa (= to bet, stávka = bet).BTW, as far as I can tell Slovakian 'štavit’ sa' means 'to bet, to gamble' (similar to Polish stawiać), so it doesn't look related to 'squeeze' to me; ...
чавити ... можливо через форму розщавити (рожчавити) внаслідок перерозкладу = ... as a result of [wrong] decomposition.
The verb šťaviť (Slovak) / šťaviti (Czech) is derived from the noun šťava/šťáva < *sъčava < *sъkěva/sъkjava, related to sok (= juice). It mostly means to worry, to bother sb (or reflexive šťaviť sa = to bother oneself). However in Czech it has several other colloquial or slang meanings.
Když jsem si koupila odšťavovač, začala jsem šťavit ve velkém. = When I bought a juice extractor, I began to squeeze [fruit] on a large scale.