There’s huge confusion between some Spanish verbs: quedar, quedarse, reunir and reunirse. The confusion is both in meaning and in construction, so we have some work to do here.
quedar ‘meet (up)’
The most common meaning of quedar is ‘meet (up)’, that is, you have agreed to see someone to do whatever. This verb is a bit informal, but acceptable in most contexts.
Note the different tenses! Though Voy a quedar… would also make sense, the commonest is the pretérito perfecto compuesto (= “I’ve agreed to meet…”).
quedarse ‘stay; remain’
When quedar is pronominal, that is, a “reflexive” pronoun is used (yo me quedo, tú te quedas, etc.), it means ‘stay; remain’.
The verb reunirse, pronominal (plain reunir [algo] means ‘gather, collect [something]’), means ‘meet’ too. The difference between reunirse and quedar is that reunirse is quite formal, while quedar is neutral/informal.
You should use reunirse when you are meeting your boss, etc. Otherwise, quedar is probably the best option.
Let me insist: reunirse is pronominal, so a “reflexive” pronoun should be used. You should say “nos reunimos”, “me reúno”, etc.
Dos and don’ts of cita
While we’re at it, let’s also see some theory about the word cita, which means both ‘date’ and ‘appointment’:
As you can see, cita is used in both contexts, let’s say “formal” and “romantic”, but it’s a bit weird (though not incorrect) in an intermediate context, let’s say “friendly”: