iptables rules break communication between Docker containers - nginx

Question

nginx-proxy is a Docker container that acts as a reverse proxy to other containers. It uses the Docker API to detect other containers and automatically proxies traffic to them.
I have a simple nginx-proxy setup: (where subdomain.example.com is replaced with my domain)
docker run -d -p 80:80 -v /var/run/docker.sock:/tmp/docker.sock:ro jwilder/nginx-proxy
docker run -e VIRTUAL_HOST=subdomain.example.com kdelfour/cloud9-docker
It works with no problem when I have my firewall off. When I have my firewall on, I get a 504 Gateway Time-out error from nginx. This means that I'm able to see nginx on port 80, but my firewall rules seem to be restricting container-to-container and/or Docker API traffic.
I created a GitHub issue, but the creator of nginx-proxy said he had never run into this issue.
These are the "firewall off" rules: (these work)
iptables -F
iptables -P INPUT ACCEPT
iptables -P FORWARD ACCEPT
iptables -P OUTPUT ACCEPT
These are my "firewall on" rules: (these don't work)
# Based on tutorial from http://www.thegeekstuff.com/scripts/iptables-rules / http://www.thegeekstuff.com/2011/06/iptables-rules-examples/
# Delete existing rules
iptables -F
# Set default chain policies
iptables -P INPUT DROP
iptables -P FORWARD DROP
iptables -P OUTPUT DROP
# Allow loopback access
iptables -A INPUT -i lo -j ACCEPT
iptables -A OUTPUT -o lo -j ACCEPT
# Allow inbound/outbound SSH
iptables -A INPUT -i eth0 -p tcp --dport 22 -m state --state NEW,ESTABLISHED -j ACCEPT
iptables -A OUTPUT -o eth0 -p tcp --sport 22 -m state --state ESTABLISHED -j ACCEPT
iptables -A OUTPUT -o eth0 -p tcp --dport 22 -m state --state NEW,ESTABLISHED -j ACCEPT
iptables -A INPUT -i eth0 -p tcp --sport 22 -m state --state ESTABLISHED -j ACCEPT
# Allow inbound/outbound HTTP
iptables -A INPUT -i eth0 -p tcp --dport 80 -m state --state NEW,ESTABLISHED -j ACCEPT
iptables -A OUTPUT -o eth0 -p tcp --sport 80 -m state --state ESTABLISHED -j ACCEPT
iptables -A OUTPUT -o eth0 -p tcp --dport 80 -m state --state NEW,ESTABLISHED -j ACCEPT
iptables -A INPUT -i eth0 -p tcp --sport 80 -m state --state ESTABLISHED -j ACCEPT
# Allow inbound/outbound HTTPS
iptables -A INPUT -i eth0 -p tcp --dport 443 -m state --state NEW,ESTABLISHED -j ACCEPT
iptables -A OUTPUT -o eth0 -p tcp --sport 443 -m state --state ESTABLISHED -j ACCEPT
iptables -A OUTPUT -o eth0 -p tcp --dport 443 -m state --state NEW,ESTABLISHED -j ACCEPT
iptables -A INPUT -i eth0 -p tcp --sport 443 -m state --state ESTABLISHED -j ACCEPT
# Ping from inside to outside
iptables -A OUTPUT -p icmp --icmp-type echo-request -j ACCEPT
iptables -A INPUT -p icmp --icmp-type echo-reply -j ACCEPT
# Ping from outside to inside
iptables -A INPUT -p icmp --icmp-type echo-request -j ACCEPT
iptables -A OUTPUT -p icmp --icmp-type echo-reply -j ACCEPT
# Allow outbound DNS
iptables -A OUTPUT -p udp -o eth0 --dport 53 -j ACCEPT
iptables -A INPUT -p udp -i eth0 --sport 53 -j ACCEPT
# Allow outbound NTP
iptables -A OUTPUT -p udp -o eth0 --dport 123 -j ACCEPT
iptables -A INPUT -p udp -i eth0 --sport 123 -j ACCEPT
# This bit is from https://blog.andyet.com/2014/09/11/docker-host-iptables-forwarding
# Docker Rules: Forward chain between docker0 and eth0.
iptables -A FORWARD -i docker0 -o eth0 -j ACCEPT
iptables -A FORWARD -i eth0 -o docker0 -j ACCEPT
ip6tables -A FORWARD -i docker0 -o eth0 -j ACCEPT
ip6tables -A FORWARD -i eth0 -o docker0 -j ACCEPT
iptables-save > /etc/network/iptables.rules
Why won't the proxy work when I have my firewall on?

Answer 1

Thanks to advice by Joel C (see the comments above), there was a problem on the FORWARD chain which I fixed like so:
iptables -A FORWARD -i docker0 -j ACCEPT