The nearby (3.2 pc) K2V star ε Eridani hosts the closest detected debris disk and an exo-Jupiter analog (ε Eridani b). In this work, we reanalyse ALMA Band 6 (1.29 mm) observations of ε Eridani to assess whether its outer debris belt exhibits a non-zero eccentricity, potentially induced by planet-disk interactions. Considering two models, one that accounts for the disk's forced eccentricity (ef) and another that accounts for a proper eccentricity (ep), we place stringent (99.7th-percentile) upper limits of ef < 2.6% and ep < 6.8%, respectively, and find tentative evidence for a non-zero forced eccentricity of ef = 1.62^(+0.39)_(-0.40)% at the 4σ level. This corresponds to the lowest millimeter-derived upper limit on the forced eccentricity of any debris disk to date, which, together with the low upper limit on the proper eccentricity, suggests that the outer belt is nearly circular and dynamically cold. We further explore planetary system architectures that are consistent with these ALMA-based results in combination with Spitzer/IRAC, JWST/NIRCam F444W, and JWST/MIRI F2550W data. Together, these analyses provide insight into potential planet-disk interactions in this nearby system, constraining any additional planet beyond ε Eridani b to masses ranging from ∼ 2×10⁻⁴ to 0.3 MJup at orbital separations of ∼ 6-60 au.