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We present the results of a high angular resolution Very Large Array (VLA) Class I 44 GHz and Class II 6.7 GHz methanol maser survey of a sample of ~20 massive young stellar object (MYSO) outflow candidates selected on the basis of extended 4.5 micron emission in Spitzer Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE) images. These 4.5 micron-selected candidates are referred to as extended green objects (EGOs), for the common coding of this band as green in three-color IRAC images. The detection rate of 6.7 GHz Class II methanol masers, which are associated exclusively with massive YSOs, towards EGOs is greater than ~64%--nearly double the detection rate of surveys using other MYSO selection criteria. The detection rate of Class I 44 GHz methanol masers, which trace molecular outflows, is ~89% towards EGOs associated with 6.7 GHz methanol masers. The two types of methanol masers exhibit different spatial distributions: 6.7 GHz masers are centrally concentrated and usually coincide with 24 micron emission, while 44 GHz masers are widely distributed and generally trace diffuse 4.5 micron features. We also present results of a complementary James Clerk Maxwell Telescope (JCMT) single-pointing molecular line survey of EGOs in the outflow tracers HCO+(3-2) and SiO(5-4). The HCO+ line profiles and high SiO detection rate (90%) are indicative of the presence of active outflows. No 44 GHz continuum emission is detected at the 5 mJy/beam (5 sigma) level towards 95% of EGOs surveyed, excluding bright ultracompact HII regions as powering sources for the 4.5 micron outflows. The results of our surveys constitute strong evidence that EGOs are young, massive YSOs, with active outflows, presumably powered by ongoing accretion.