Few fonts include these combinations. As of this writing, the fonts I know of that have epsilon and omicron with a circumflex: the more recent versions of the GreekKeys fonts (the free-to-all New Athena Unicode, plus AttikaU, BosporusU, and KadmosU, which are part of the GreekKeys package), IFAO-Grec, and the fonts associated with Antioch (Vusillus, Hyle, and Orthos).
See the end of this post for more information on how to obtain these fonts. And note that since the fonts can only be installed on desktop/laptop computers, I know of no way to use these characters in iOS.
Input using GreekKeys 2015
If you have the GreekKeys 2015 input method enabled, these characters can be typed like any other circumflex-over-vowel combination (namely option-3, followed by the vowel). This works even if you are using IFAO-Grec or Vusillus, rather than the GreekKeys fonts.
As far as I know, only the GreekKeys fonts have a character that combines both a macron and circumflex over either epsilon or omicron. This is typed: option-3, option-+ (i.e. option-shift-=), then the vowel. See the featured image above or download the sampler pdf to see how these look.
Other input methods
The Unicode Standard has not yet designated code points for epsilon with circumflex or omicron with circumflex. But the designers of the fonts above have all agreed to use the same code points in the “Private Use Area”: epsilon with circumflex is at the Unicode code point E1B0 and omicron with circumflex is at E1C0. Knowing these code points allows you to enter the characters using Unicode Hex Input.
You can also search for these code points in the Mac Character Viewer and save them to your Favorites. Unfortunately, the increasing suckiness of the Character Viewer makes this a lot less useful than it once was. Compare how one of those Private Use Area code points looked in the Character Viewer five years ago (perhaps under OS 10.8) …
with how it it looks under Yosemite
In the Yosemite version of the Character Viewer, Private Use Area characters are not shown, even if you have a font installed that includes the character. Moreover, the normal Font Variation section does not appear, so that you cannot tell whether you have an appropriate font installed or not; notice how in the earlier version of the Character Viewer, I could easily see all the fonts that I had installed that had an epsilon with circumflex at that code point, as well as what other fonts used that code point for different characters. Nevertheless, if you can remember which unhelpful square in your favorites is which, you can still double-click the square to enter the character.
You can see what these characters look like in the various fonts in this pdf.
Font and input download information
IFAO-Grec may be downloaded from http://www.ifao.egnet.net/publications/publier/outils-ed/polices/ (scroll down to the Download section where you see something like “IFAO-Grec Unicode” and a link to a zip file.)
If you know of other fonts that support these characters, please leave a comment.