Implicit neural representations (INRs) have recently achieved impressive results in image representation. This work explores the uncertainty quantification quality of INRs for medical imaging. We propose the first uncertainty aware, end-to-end INR architecture for computed tomography (CT) image reconstruction. Four established neural network uncertainty quantification techniques -- deep ensembles, Monte Carlo dropout, Bayes-by-backpropagation, and Hamiltonian Monte Carlo -- are implemented and assessed according to both image reconstruction quality and model calibration. We find that these INRs outperform traditional medical image reconstruction algorithms according to predictive accuracy; deep ensembles of Monte Carlo dropout base-learners achieve the best image reconstruction and model calibration among the techniques tested; activation function and random Fourier feature embedding frequency have large effects on model performance; and Bayes-by-backpropogation is ill-suited for sampling from the INR posterior distributions. Preliminary results further indicate that, with adequate tuning, Hamiltonian Monte Carlo may outperform Monte Carlo dropout deep ensembles.