The issues about nitrogen fixation and essential amino acids are separate issues. Essential amino acids are not essential for their nitrogen content; rather, they are essential because animals lack the enzymes to build the required carbon skeletons.
Many amino acids are synthesized from other amino acids or from intermediates of glucose metabolism. For example, in most organisms, lysine, methionine, and threonine are synthesized from aspartate (which itself can by synthesized from glucose). The first step in this process is catalyzed by an enzyme called
aspartokinase, which animals lack. Thus, whereas most other species (plants, bacteria, fungi, etc.) can convert sugars into aspartate into lys, met, and thr, animals are incapable of performing these chemical reactions and thus must obtain these three amino acids from their diet. This applies to the other essential amino acids as well.
Thus, good evidence for the essential amino acids comes from the human genome itself. We know what enzymes are required to synthesize each amino acid, and we can see which of these enzymes are encoded in the human genome. If the human genome lacks many of the enzymes critical for the synthesis of a particular amino acid, the amino acid must be obtained in our diet (the fact that these gene losses are evolutionarily conserved among animals adds further to this point).
With regard to nitrogen fixation, we know what enzymes are involved in nitrogen fixation in bacteria, and humans lack enzymes that resemble these. We have sequenced the content of our gut microbiome and, although I have not looked at this data in detail, I do not think it contains nitrogen fixing bacteria. You may be interested in reading the following paper (which I found while browsing through the Voet & Voet chapter on amino acid metabolism for this post), which argues that amino acids are the primary source of nitrogen for animals:
Katagiri and Nakamura. 2002. Animals are dependent on preformed alpha-amino nitrogen as an essential nutrient. Life 53:125.
doi:10.1080/15216540211467