I can get a new Recruit assigned to my Ship but he is listed separately from my existing 5-man Squad.
Even if I Dismiss a member of my existing Squad (making a 4-man Squad), the new Recruit is not assigned to my Squad.
How do I get the new Recruit into my 4-man Squad to make it a 5-man Squad?
Recruits are assigned to a base. One must go to that base to get them into the squad. As long as there is enough room on the aircraft, they can be selected to enter the craft and join the squad.
I know that, and that is what I do.
You fail to give step-by-step on HOW to assign a Recruit to my Squad.
Like how to change my 4-man Squad to a 5-man Squad by assigning the Recruit. The Recruit IS listed but separately from my Squad.
Exactly where is the recruit?
your recruit is probably at one of the bases. You need your aircraft stationed there to be able to add that recruit to your other soldiers on the aircraft. When your aircraft will be there then:
in Personnel screen → use black icon with some number and with dropdown list → and change from “Name of the base” to <Manticore #> or other airplane that you use. Next to airplane name will be indicator how many soldiers are in there, for example 4/6, which means that two soldiers can go in there. Worth remembering is that vehicles and Mutogs take 3 spaces instead of 1. So if it is a Mutog you want to put in your airplane then you need to make 3 free seats in the aircraft.
I wouldn’t say that this is the mechanic that should cause any trouble… So you will probably have some other questions soon.
Finally got Recruits to join my Squad.
I’m an XCOM player and since your ship alway flew back to your base, handling Recruits was easier.
Phoenix Point, of course, the Recruits have to be located at the base where your Ship is located.
Thanks to all who replied. 
Some of what many see as no issues are not always so easy for others to get their head around them. Glad you got this one sussed out.