I am thinking of just attaching a big cap on U1 to try to dampen this out but this is really just a guess
Getting the Vcc straight is always a good idea. A big cap alone will not help. To cover a broad range of frequencies you need to use capacitors with different resonant frequencies in parallel. A typical choice could consist of a 1µF tantalum + 100nF ceramic + 100pF ceramic (provide some pads of suitable geometries so you can try different combinations in the layout). Remember to use ultra-short connections between the IC and the capacitors and from the capacitors to GND.
You may also consider inserting a ferrite in series with Vcc to further dampen high frequency noise.
Another measure is a GND plane underneath the traces from the IC to the ETH-transformer. You may also want to place the IC as near as possible to the transformer to keep the trace lengths very short.
Also note that you have GND and EARTH in your design. These are ideally DC-connected only at a single point in the design, typically somewhere in the power supply section. Make sure that you do not create ground loops by having more than one connection between GND and EARTH. Having an AC connection by a small capacitor to divert HF-noise to EARTH is allowed at more than one point, although the number of these connections should be kept as small as possible.
You should also investigate the quality of your DC/DC converter's output. This type of converter is especially sensitive to layout and can create lots of noise if the layout is unsuitable. Preventing switching noise from being generated in the first place is the best option - much better than trying to suppress it later on.
Here is some background information on reducing radiated emissions in Ethernet systems