Does an external magnetic field help to obtain a BEC at higher temperatures than without it?

I feel you are missing a key point of a BEC, which is that the particles should ideally be non-interacting. Using statistical arguments the original papers show that the ground-state becomes macroscopically occupied at $T_c$. This idea is so appealing, because it is extremely simple and fundamental. I believe high school students are able to understand this

The idea of entanglement due to the interaction between magnetic moments does not only differ in many key aspects from the original proposed BEC, it is also much more complicated. Therefore, even if one could prove that entanglement is equivalent to BEC (which is not the case) I don't see the point of using this perspective.

f one takes the magnetic dipole of the electrons and not the spin as the starting point for explaining the phenomenon of Bose/Einstein condensates

No. Bose-Einstein condensation is driven by quantum statistics, not by any interactions and hence not by magnetic dipole or spins.

a BEC comes about through the undisturbed alignment of the magnetic dipoles of the electrons

No.

You can also have coexisting BECs of different spins or dipole moments, that is called a fragmented BEC. Interactions among these would probably cause problems though.

Does an external magnetic field help to obtain a BEC at higher temperatures than without it?

Not because of what you are thinking.
But if the external field changed the trap so that it is tighter, then yes it may affect $T_{text{c}}$.