There would be a huge problem with:
1. Subc. R. being another big poor region with bad infrastructure
2. Another mass of unemployed, weighing on social system and finance of the state
3. cancelling current Schengen border and rebuilding a new one (costs much)

It would be no win for them, too:
1. Currently they are a small forgotten region "somewhere in the far west" of Ukraine, after joining Slovakia they would be the same small forgotten region "in the far east".
2. Culturally, ukrainization would be weakened, true, but replaced by slovakization. Education and culture in Slovakia is in terrible state, everything is slovakized, or even better, "west-slovakized" – even culture, language and history of certain parts of Slovakia are omitted – why, I only suspect…
3. They would not have any sort of autonomy, please, Slovak nationalists get rabid when hearing the word "autonomy".
edit – 4. Back to slovakization:  some Slovak nationalists say that Subc. R. is a stolen part of Slovakia. And not to say that many people in the western or central parts of the country are confused about Rusyns, Ukrainians, Russians. For them anything written in Cyrillic is Russian, Ukrainian = Russian = Rusyn, somewhere in the same area as Belorussian, Romanian…  ;)