Maybe you're thinking of Cold War stuff? However, Sweden has a longstanding policy of political neutrality (not unlike Switzerland, but for different reasons), so, it would have to be Norway that builds the icebreakers.
Swede, here. Nah, Sweden's neutrality only extends as far as keeping itself alliance free (and it is questionable whether it can be called even that for the last 70 or so years). Sweden actually has a pretty huge weapon industry (third biggest in the world in arms exports per capita, iirc), in the spirit and tradition of Alfred Nobel - yes, the one with the Nobel Prize - who invented dynamite and smoke-free blackpowder or whatever and created the Bofors weapon industry giant. Remember the number one pro of not being tangled up in any alliance or league - you can trade with both sides. Profit!
I'm pretty sure that we're required by law to build all of out military equipment in the US proper. A lot of our stuff is still produced by non-American companies, but the factories are physically located in the US, so outsourcing wouldn't really help here.
After mulling it over some in my head, I think what I was remembering actually was weapon research rather than weapon production, so yeah, that's very possible.
the President didn't believe the arctic was warming and thus no icebreakers are needed
I get the rest of your point, I just felt the need to point out how beautifully self-contradictive this issue sounds if you remove the context. "The arctic ice is melting, so we need more arctic icebreakers!"