Four Swiss players – Admir Mehmedi, Breel Embolo, Blerim Dzemaili and Granit Xhaka – had to change their shirts in the first half of the goalless draw in Lille on Sunday.
Recent Arsenal recruit Xhaka had to change his shirt again in the second half.
"I hope Puma doesn't make condoms," Shaqiri told Switzerland's SRF broadcaster.
Former England striker Gary Lineker took the opportunity to poke fun on Twitter, writing: "Swiss team's Puma shirts tearing like paper. Adidas ball bursting. You can never rely on German efficiency!"
Other Twitter users compared the shirts to Swiss cheese – full of holes.
German manufacturer Puma apologized.
"Our five Puma teams have played 10 games at Euro 2016 before this without experiencing this problem," the company said.
It later blamed a manufacturing flaw. Its investigation had found that "there was a faulty batch of material in which the yarn was damaged during production", the company said in a statement.
The Swiss shirts will be needed for at least one more match as the team has qualified for the second round.
Puma's German rival Adidas said Monday it had agreed a four-year deal to remain kit supplier to Germany until 2022 worth more than 50 million euros ($56.5 million) a year.
The agreement with the world champions came in the face of competition from US giant Nike for the prestigious contract.