What happens if I push another mech with skewer?
Here is the text on Skewer:

SKEWER
[1] Move forward 1 space.
[2] Move forward 2 spaces.
[3] Move forward 3 spaces.
If Skewer kills a Minion, place it on this card (max of 1). Before drawing damage, you may instead scrap the Minion.

Here are the rules for &quot;stomp&quot; (under the Minions section):

Minions will also die if a Mech stomps on them (moves
into the same space).

And, finally, the relevant rules for &quot;Push&quot;:

Player Mechs can Push [..] other player Mechs [...].
[...]
Executing a Move Command that would put the Mech into a space containing another Mech [...] pushes that object forward in the same direction, which could push another Mech [...].

Now that we have the rules established, here is your first question:

If my Mech has a Skewer card and I push someone else into a Mech do I get the minion on my card?

In light of the cited rules, the answer is... unclear. The answer depends on your interpretation of what exact event is taking place when the minion is killed.
That is, is the stomp a sub-event within Skewer or is stomp a result of Skewer? If your interpretation is the former, then yes the minion would be placed on the card. If your interpretation is the latter, then no.
At this time, there is no official clarification that distinguishes the granularity of stomp within the context if its causal action that could be used to resolve this ambiguity. My personal interpretation is the Skewer is triggering stomp and thus the minion is not killed via Skewer directly. This interpretation also makes more thematic sense as the pushed Mech does not necessary have Skewer equipped (and thus it has nothing to physically skewer with!).
We do have enough information to firmly answer your second question, however:

If another player has Skewer and I push them into a minion do they get the minion?

The answer is no, they do not gain the minion. The card specifically says &quot;If Skewer kills a Minion&quot;.
Killing a minion any other way does not trigger the ability. If the card could be triggered by other means, the text would have been written as such (e.g., &quot;When your Mech kills a minion...&quot;).
Regardless of the thematic implications, the rules are clear.
Source: Printed rules (rulebook and card)