Modern ballistic missiles generally have solid fuel stages (for quicker preparation and more rugged handling) rather than liquid. So it is more than a coincidence that the first, second and third stages of the Epsilon are solid fuel.
The Epsilon's specifications are very similar to the fully developed (but then cancelled) US MX ICBM. For comparison the MX's specs are Height 22m, Diameter 2.3m, Weight 97 tons, 3 stages, blast yield 3 Mt total (using up to 10 MIRVs).
The extent to which the US assisted JAXA's Epsilon Project is unclear. Japan Aerospace eXploration Agency (JAXA) is Japan’s NASA equivalent. Space agencies have dual military-civilian use technology and dual-use career personnel. JAXA, and of course NASA, are dual-use - although space agencies rarely admit this.
NTI August 23, 2013 reported on Japan's increasing ability to develop an ICBM - which of course are the primary means of delivering nuclear weapons http://www.nti.org/gsn/article/japans-new-military-buildup-seen-response-north-korea-china/ :
"Japan's New Military Buildup Seen as Response to North Korea, China"
After decades of hewing to a strictly self-defensive military posture, Japan in recent months has indicated it plans to acquire offensive military capabilities such as ballistic missiles that could be used to carry out advance attacks on North Korea's strategic assets.
Some of Japan's space-program activities have applications in the development of strategic weapons. On Tuesday, the island nation is slated to fire its solid-fueled Epsilon rocket which could potentially be adapted to power an ICBM.
[The first launch of Epsilon, of a small scientific satellite SPRINT-A, occurred on September 14, 2013 at Uchinoura Space Center using a two (solid fuel) stage version of Epsilon.]
These armament plans have raised regional concerns that Tokyo may be shedding its post-World War II pacifist defense posture.
"What is worrisomely ... is that Japan's rearmament would be met with China's reaction, which could cause regional instability," Korea National Defense University Japan researcher Park Young-june said.
The United States, however, is seen as supportive of Japan taking on a more assertive regional role, as it could be useful in meeting the challenge of China's growing military might."