Entweder is not a conjunction but an adverb. If an adverb takes the first position in a V2 sentence (i.e. in a main clause), per definition the verb follows on the second position and the other parts come thereafter. If you transform a standalone V2 sentence into an entweder ... oder sentence, the former part on position 1 (before the verb) is moved behind the verb that takes the second position.
Take this as an example for the basic principle of moving around the first part of a sentence in a V2 sentence:
Ich gehe sonntags gerne spazieren.
Sonntags gehe ich gerne spazieren
Now we make a similar sentence that features an alternative with oder:
Ich gehe spazieren oder [ich] schaue mir einen Film an.
The second ich can be omited in the form of an ellipsis.
We will now see that entweder is a normal adverb and can be shifted around like sonntags in the example above:
Ich gehe entweder spazieren oder [ich] schaue mir einen Film an.
Entweder gehe ich spazieren oder [ich] schaue mir einen Film an.
Your example Entweder du bist Teil der Lösung, oder du bist Teil des Problems violates the fundamental definition of a V2 sentence because there would be two sentence parts in the Vorfeld (i.e. before the verb), making it actually an ungrammatical V3 sentence. (However, in spoken language there might be situations where native speaker move entweder to a 'Vor-Vorfeld', but this would be considered non-standard and mostly caused by psycholinguistical constraints).
The second sentence (after oder) remains unchanged, however, because oder, unlike entweder, is in fact a conjunction.