To my mind, though it's not false, it's not true, and anyone who claims it is without understanding why should be examined closely for signs of snake oil.
Way back in the early days, firewalls (when they existed) were simple packet-inspection devices. They'd make a decision about whether to forward or drop a packet based solely on characteristics of the packet itself. You might say "that packet comes from TCP port 80, it has the SYN and ACK flags set, so it is (most likely) a web server's response to an initial TCP connection attempt, so let it in.
A significant improvement in firewall security was the stateful firewall. This had a memory of traffic flows through it prior the to consideration of any given packet that was used to inform the decision about that packet.
With a stateful firewall, you could instead say "that packet comes from TCP port 80, and the source address and port match the destination address and port of a TCP SYN packet that went through me about 75ms ago, so it is a web server's response to an initial TCP connection attempt that started inside my trusted network, so let it in. Or you could look at an ICMP echo-reply, and permit it only if it corresponded to a recent echo-request to a particular external server. And so on.
The thing about NATting firewalls is they are implicitly stateful. Because all traffic from inside to out must be rebadged with a new source address before the internet has any chance of replying to it, traffic can only flow from outside to inside if the firewall has a record of which particular internal machine recently sent out some traffic to generate the reply. If the firewall has no such record, it cannot rewrite the destination IP address of the inbound packet, because it won't have the faintest idea which internal address to rewrite it to.
So my feeling is that stateful firewalls are more secure than stateless firewalls, and all NATting firewalls are implicitly stateful. Of course, every other firewall I've seen in the last decade is statefully-capable as well, so as long as a non-NATting firewall wasn't very badly configured, it, too, offered similar security. I have never seen any analysis that suggested to me that NAT had any security advantages other than implicit statefulness.