To make sure if you can use अपना (or one of its variants, अपने and अपनी) in a sentence, or more precisely, a clause, place अपना where you want to, and then check if it comes after the subject (or all the subjects, if there are multiple subjects) of the clause. If it does, you can use अपना; if it doesn't, you can't. Let me now explain this in the context of the given sentence.
The sentence "नेहा और (उसके/अपने) दोस्त जा रहे हैं" contains two subjects (नेहा and दोस्त), and only one is preceded by अपने. Hence, अपने cannot be used.
But we can modify the sentence a bit so as to have just one subject: नेहा X दोस्तों के साथ जा रही है। (Neha is going with her friends). X is where we are going to see if अपने works. Since the sole subject of the sentence is नेहा, and since it comes before X, अपने can be used.
Glad to know it was helpful. That's definitely one way of looking at it, but it may not always work. I gave some more thought to this topic after posting my first comment and decided to phrase things in a slightly different way this time.
Basically, अपना has to be outside of what constitutes the noun phrase of the clause; having conjunctions inside the noun phrase as long as long as that condition is met shouldn't be a problem.
So, in नेहा और अपने दोस्त जा रहे हैं (noun phrase in italics), अपने is part of the noun phrase, and hence the construction isn't correct.
However, a sentence like नेहा और उसके दोस्त अपने स्कूल जा रहे हैं, (Neha and her friends are going to their school) has "अपने" outside the noun phrase (which also happens to have a conjunction), and hence is perfectly valid.
Finally, one more example with a relatively long noun phrase:
यहाँ रहने वाले और उनके दोस्त कल अपने बच्चों से मिल सकते हैं। - Those who stay here and their friends can meet with their children tomorrow.
@Zesul - thanks a lot for clear explanation! But I'm wondering why is it "friends" and "friend" is wrong? Is it cuz of उसके that marks it as a plural form? Or?
Is it also grammatically correct to say "नेहा उसके दोस्त के साथ जा रहे हैं।"?
In this case, the verb should agree with just 'Neha' (because the subject of the sentence is just नेहा) so it should be नेहा उसके दोस्त के साथ जा रही है। (instead of जा रहे हैं ).
Also, it means "Neha is going with her friend" (singular friend). Because of the postposition के, 'उसके दोस्त' is in the oblique case. If you want to say "Neha is going with her friends" instead, you should use the oblique form of the plural दोस्त which is दोस्तों. So, the sentence would be नेहा उसके दोस्तों के साथ जा रही है।
It's because the sentence uses उसके instead of उसका indicating that दोस्त is meant to be plural.