No. The Pi Zero uses the BCM2835 system-on-a-chip, which combines a CPU and a VideoCore 4 GPU -- the same basic SoC as on the Pi A/B/+ models although with a faster clock speed (which does not necessarily mean it was manufactured any differently1).
My understanding is that the GPU bootstraps the CPU and loads a kernel into it. Although the kernel can be anything, including linux, which in its vanilla form meets the FSF's standards for free/libre open source software, the firmware required for the GPU is from closed source.
I'm pretty sure the non-vanilla Raspberry Pi kernel, which presumably runs on the Zero, does qualify as FOSS. The bits added for the BCM2708 (of which the 2835 is an implementation) are not proprietary. In other words, the fact that this has not been merged into the vanilla kernel is not a licensing issue.
So, this is the same in this regard as previous Pi's and nearly all general purpose computers currently on the market, which use proprietary firmware for the BIOS or (U)EFI. Looking at the article you linked, the issue with other single board computers seems to most often be firmware for peripheral components (wifi, VPU/GPU, etc).
1. I'd guess it was not and this in part reflects much successful overclocking on the pi -- congratulations gang!