I have never taken Eltroxin (it isn't available in the US, officially), but can offer this insight:
All Synthetic thyroid meds (Eltroxin, Levothroid, Levoxyl, Levo-T, Synthroid, Unithroid, etc.) contain the same active ingredient, Levothyroxine Sodium.
The difference between the brands is in the fillers that are used to give the pills their shape, size, color, and texture. The differences in fillers can affect the way you absorb the active part of the pills. One person may absorb Synthroid better than Eltroxin, but the next person may absorb the Eltroxin better.
The best advice I can offer you is that if you DO switch brands, then get labs done 6-8 weeks after the switch and adjust your dose if necessary at that time.
I am not a doctor, nor have I ever played one on TV...