Replacing Interlocking Ceiling Tiles From The "Wrong" Direction
Checking the Armstrong web site from the link you left they state that these particular tiles should NOT be stapled (for whatever reasons) and the warranty will be voided if done so.
There is also a question asked in their FAQ's asking this same problem :&quot;can tiles be stapled?&quot;  The rep answers with a curt yes and continues to reiterate the warranty void issue. That's all that's said.
I would try to remove the staples from the tiles that border the damaged area. Since that's what is preventing the new ones from being installed. Once the first new tiles are installed the remaining new ones will go in as per manufacturer's instruction.
As for method of securing them; the company states glue (like the Henry's you purchased) or their &quot;ceiling track&quot; (which doesn't work with your existing tiles). The &quot;Accusto-Gum&quot; adhesive is recommended for plaster and drywall surfaces. You may want to consider using an adhesive with more holding power. Any construction adhesive should hold. Staples or Henry's adhesive both are not advised for your repair. The staples have held for how long? Use both for increased holding power.