Block Instagram for Android to access the Internet, using a software like SQUID on my Laptop/Desktop
If you have access to the WiFi Router you can block traffic and/or mac address. there are custom firmware for some router models like DD-WRT or Open-WRT. But the phone still can by-pass via mobile data.

Filter rules are complicated and may have unwanted effects. This is just a example

"Instagram" uses TCP protocol for communication.

TCP traffic analysis

Ports Range

Remote tcp port(s): 443, 80.

HTTP traffic analysis

HTTP traffic is found in Instagram's traffic.

Host(s) being connected(regular expression): ^..ak.instagram.com, ^..cdninstagram.com.

TLS traffic analysis

TLS connections are found in Instagram's traffic.

TLS Host(s) being connected(regular expression): instagram.com.

How to block Instagram?

Block the listed ports to block Instagram.

Block the listed http and tls domains to block Instagram.

Setup a content filter(ie: WFilter), which can block Instagram via signature

http://imfirewall.us/protocols/view.php?proto=Instagram