You are right to be concerned, eucalyptus is prone to sprout from the stump and also the roots.
The best way to stop regrowth, I think is to drill several holes around the edge of the stumps surface, go down a couple of inches, if you can, but an inch should do.
Fill the holes with a proprietary, glyphosate based stump killer, like 'Stumpout' or 'Brushwood killer'. Be sure to plug the holes with clay or suchlike, to stop the rain splashing it out and to stop it getting in to the environment, and for an extra safety precaution cover with plastic sheeting and secure with string.
More detailed instructions will be found on the pack.
One of the most funny you-tube video's I've seen was where a guy tried to pull out a stump with a rope attached to the tow bar of his car and all of a sudden, the stump came out and went through the rear window with great speed! job done. ;-)