Hmm. They might turn to mush. I'm not sure. Have you thought about pickling them?

I might can more mature West India burr gherkins (due to their cooked texture), but I'd be worried about a regular cucumber. I understand that some people do heat-can cucumber pickles sometimes, but I have yet to know much about how it turns out for them.

I don't know if this is for heat-canned pickles, but I hear using the cucumbers right away (as soon as you harvest them) helps them to stay crunchier. Grape leaves are also supposed to help (due to the tannins in them).

I know someone who pickles without heating them (but manages to seal the jars, somehow). I've never done that, but I've lacto-fermented using vinegar before, and that gave me ideas as to how it might work. Adding vinegar to a ferment can cause the jar to suck the lid down (I assume it could cause a jar with a canning lid to seal). Then, due to the vinegar and salt content, and cool storage, it could ferment very slowly for a long time (and you could just consider the fermentation period its unrefrigerated shelf-life). It sounds doable, anyhow. As to why the lid sucks down, I think that's because of acetic acid bacteria using up the oxygen in the jar (I don't know if that's the case, but it sounds possible, but it happens with distilled white vinegar—not just those with mother). Disclaimer: I don't claim this is safe, but I personally know of no reason why it wouldn't be.

My dad grew lemon cukes years ago in addition to regular ones. My mom pickled both, usually in the same jar. The lemon ones were always mushy while the regular cukes were still crisp. I don't know if they were over-ripe or if they just had more water in them, but they didn't work out well for us.

I have made them into refrigerator pickles several times and that worked well.