Sorry for the long post, but this is one way to do it if you want the right tools for the job. Should cost about $10, and you'll do the maintenance more often if its easy.
The right punch for getting them started is this one, or the equivalent. Even made in the USA!
7/32" hardened steel pin punch - Starrett Model #990-0184
Price: $5.40 ea
From Enco online: https://www.use-enco.com/CGI/INSRIT?PARTPG=INLMKD&PMPXNO=3009579&PMAKA=990-0184
A 3/16" punch is OK, but 7/32" is ideal.
Take a 1/4" drill or reamer and run it lightly into the ends of the holes, just to remove the indents that keep the rod from backing out on its own (never saw that happen). Squirt it down good with penetrating oil for a few days beforehand, if you can. Kroil is the best, but WD40 will probably work. By the way, a can of Kroil is a very good thing for jet outboard owners to have around. (https://www.kanolabs.com/)
Tap the punch just enough to get the rod moving a little, then tap it the other direction to distribute the P-oil. Use a metal hammer, not a soft one. You need the shock factor of a real manly hammer. Keep tapping the rod back and forth until it moves easily. Keep adding P-oil to all the slots as you go. You shouldn't have to hit it very hard, if you use a hardened punch and a metal hammer. If you use a soft piece of steel or screwdriver, etc., you'll just bend or mushroom the tip of the tool. If that happens, you might get the punch stuck too.
Once its loose enough to move, tap it as far as you can with the pin punch. Then take a piece of 3/16" drill rod an inch or so longer than the housing, and tap the rod the rest of the way out. You can get a piece of drill rod at any metals supply house for a few bucks. Pins are made to drive in and out. If you grab them with vise-grips, you'll probably nick or gouge them up, then they won't go in or out nice.
If the rod is bent or buggered up, I'd replace it. They're supposed to be straight, not sprung. Grease it up good before you re-install it. After you re-install the rod, use a center punch or chisel to put new LIGHT indentations around the holes on both ends, to replace the ones you drilled out.