Are there any cards exclusive to the Rey or Kylo Ren starter packs, or can I get all the cards from boosters if I just buy the 2 player box?

There are a few exclusive to each starter pack and the 2 player box. You need to buy at least 1 of each.

Sadly the starter sets do have some good exclusive cards in them. While you don't need them to play, if you wanted a play set, you would need to two of each which puts you in for $120. This is a CCG though and it is flat out not cheap to play at a competitive level.

If you just wanted a small simple card/dice game to play for fun, then buying two of the new 2 player box and calling it a day works pretty good actually.

The sold-separately Kylo and Rey starters are a part of the Awakenings set. Fourteen of those cards are exclusive to the starter decks.

The cards in the new two-player starter box are labeled/numbered as if they were their own set but there are a number of cards in the box from the Awakenings and Spirit of Rebellion sets.

Buying one copy of each starter will give you most of the fixed cards you need. Kylo and Rey come with two copies in each starter, and there are some cards shared between the starter so you end up with a playset (e.g. Lightsabers).

The exceptions to this are Finn, Mind Probe and Force Throw. Finn is bad, but Mind Probe and Force Throw are very powerful cards. However there are other alternatives to them at this stage, and we'll likely see more options that can replace them in future sets.