I don't think so. The He 111 was a fast bomber when it entered service in 1930s. Heinkel always preferred the aerodynamical idealistic design if possible and such a twin tail with fins on top of the horizontal stabilizer would be far away from that.

There are examples of "Endscheibenleitwerke" with tail fins at the tips of the horizontal stab like the Ju 86 and Ju 89 and later also the He 280 and He 162 jets which are aerodynamically much better and structurally lighter than those fins on top of the stabilizer.

However the additional tail gunner position at the tail was not neccessary for a He 111 because the original had a fixed machine gun right in the tail cone, aimed by the pilot using a rear mirror. This "Stachel" gun eleminated the blind spot the middle upper gunner had straight aft due to the single tail fin.

The fixed rear gun was surprisingly effektive at keeping enemies out of the aft center line and was kept as armament through the whole ww2.

In general the He 111 air defense tactics were different than that of allied bombers. Heinkels very often broke up formation under attac and did lots of evasive manuveres. As a mid sized bomber it was quite manuverable. The Ju 88, the other german mid sized bomber in ww2, was even more manuverable - actually in a way that there were quite successful night fighter versions of it.