no you cannot destroy them

as far as getting them you need to get the approprate ship (TEC, Advent-Frigate Colony Ship/Vasari-Scout Ship)

bring that ship to said gravwell (UnColonizable GravWell - UCGW) and let them work, the ability will operate on its own-autocasts

Neutral extractors (usually known as "neutrals") are perhaps one of the most valuable assets you can have in the game, especially in the early stages. These extractors operate independent of alleigance (the rate they provide resources is the same as a built extractor at 133% alleigance) so neutrals 10 jumps away from your HW will provide the same extraction rate as neutrals 1 jump away. These guys also can support refinery ships from up to 4 refineries, compared to the 3 a built extractor can support.

Do take note, however, that enemy players can take these mines away from you after you capture them (the Vasari do this the easiest, since their scouts move about quickly), though the AI in general tends to ignore them.

For the record I liked that. It made it so that people actually had to defend the neutrals. If I could force the devs to bring back one thing of old, it would be that. It would bring a semblance of balance to neutrals.

If they could be scuttled, perhaps they should regenerate every five or ten minutes or so.  Then it wouldn't have been as bad or as big of an advantage for a Vasari player.