Glutamine, an amino acid, has been cited in several studies dealing with CFS as an abnormality. For example, “zinc + glutamine.. was associated with highly significant reductions in initially increased IgM-mediated autoimmune responses”  and many studies for IBS.
- “Patients with higher Fibromyalgia Impact Questionnaire (FIQ) scores showed increased levels of glutamine …,” 
- “Fibromyalgia patients exhibited a differential expression of 421 genes (p<0.001), several relevant to pathways for pain processing, such as glutamine/glutamate signaling and axonal development.” 
- “In comparison with healthy controls, patients with FM showed higher levels of glutamate/glutamine (Glx) compounds”  i.e. more glutamate than glutamine..
- ” Our results showed a significant reduction of glutamine (P=0.002) and ornithine (P<0.05) in the blood of the CFS samples." 
- Reduced hippocampal glutamate-glutamine levels in irritable bowel syndrome: preliminary findings using magnetic resonance spectroscopy.
Dietary L-glutamine supplementation modulates microbial community and activates innate immunity in the mouse intestine .
- “In the jejunum, glutamine supplementation decreased the abundance of Firmicutes, while increased mRNA levels for antibacterial substances in association with the activation of NF-κB and PI3K-Akt pathways.”
- “In the ileum, glutamine supplementation induced a shift in the Firmicutes:Bacteroidetes ratio in favor of Bacteroidetes, and enhanced mRNA levels for Tlr4, pro-inflammatory cytokines, and antibacterial substances participating in NF-κB and JNK signaling pathways. These results indicate that the effects of glutamine on the intestine vary with its segments and compartments.”
This effect is not desired:
- “in the stool samples there was a higher relative abundance of Bacteroidetes and lower abundance of Firmicutes observed in ME/CFS patients compared to healthy controls.” 
- “IBD patients have increased bacteroides, adherent or invasive Escherichia coli, and enterococci, and reduced Bifidobacteria and Lactobacillus species”.[Medscape]
I was unable to find any published studies on glutamine supplementation of CFS/FM/IBS. In 2010, there was trial sponsored by NIH (link), but no apparent publication resulted (which implies no benefit was found, and could imply that it may have had a negative effect) – with 2015 status stating no study result posted.
The available evidence suggests that glutamine supplementation may worsen the shift of bacteria seen in CFS/FM/IBS. The absence of study results being posted suggests there were no positive results seen. In FM, higher levels are associated with worst symptoms. Conclusion: do not supplement with glutamine.
For early notes on other amino acid supplementation, see this post.
NOTE: if you have uBiome results, supplementing with glutamine may be good for some bacteria genus overgrowths. See this post . For example: Dorea, Pseudobutyribibrio and Oscillospira overgrowths