Are there any intersections between the biochemical pathways of uracil biosynthesis (or metabolism) and methionine degradation in eukaryotes?
No, I don't think there are any intersections, and I can't see how uracil metabolism would have any influence upon methionine degradation.

Methionine degradation is shown in Figure 1 below. The end product is succinyl CoA which is fed into the TCA cycle. The only obvious by-product is cysteine which doesn't have any connection with uracil biosynthesis.

Uracil biosynthesis (as UMP) is shown in Figure 2: it is built from CO2 and NH3. Uracil degradation produces CO2 NH3 and aspartate.

There are certainly no shared intermediates in these pathways, and I can't come up with any convincing indirect connections either. 

Figure 1. Methionine degradation


Figure 2. Uracil biosynthesis