I have 6 tiles at the bottom of my shower stall that the grout has come loose. The home is only 6 years old and the caulk is still in good shape.
When the shower head is turned toward the tiles with the loose grout, the water leaks down to the basement.
When the shower head is turned away from that area there is no leaking of water.
Can the looe grout cause water to stream into the basement and how do I insert more grout to eliminate this problem?

I am no expect at this but here goes. You stated that the six tiles are located on the bottom of the shower. Even if the grout was broken I would guess that drain pan/rubber liner would still catch the water and feed it into the drain.
To fix the grout... get a grout removing tool. You can buy a manual one or I prefer one that connects to a Dremmel. Remove the grout. Clean your area and regrout it. After it is dry, 24 hours, buy some "Penetrating Sealer" for grout. You can buy this at any home improvement store. Hope this helps. I am sure one of the experts here will tell you more with details.

It sounds like when the shower was built, they did not run the sub straite over lip of shower base.

If u have a preformed base thisis the problem.

Remove 2 lines of tile and backer board.
Install wonder board.Before installing wonder board put a bead of silicone on pan lip and press wonder board into it.

Put bead of silicone were board meets base and seal board where the cuts are with tape and thin set.

Let silicone cure for 24 hours

Install tile with thin set and grout.

Good luck

Chuck