adriculous This must be your problem. You need to add ^ in front of every dependency (except zend because that's the only one that has to stay at a precise version).
I managed to run composer install with the following: (updated from your copy-paste above)
{
"name": "flarum/flarum",
"description": "Delightfully simple forum software.",
"type": "project",
"keywords": [
"forum",
"discussion"
],
"homepage": "http://flarum.org",
"license": "MIT",
"authors": [
{
"name": "Toby Zerner",
"email": "toby.zerner@gmail.com"
},
{
"name": "Franz Liedke",
"email": "franz@develophp.org"
}
],
"support": {
"issues": "https://github.com/flarum/core/issues",
"source": "https://github.com/flarum/flarum",
"docs": "http://flarum.org/docs"
},
"require": {
"flarum/core": "^0.1.0",
"flarum/flarum-ext-akismet": "^0.1.0",
"flarum/flarum-ext-approval": "^0.1.0",
"flarum/flarum-ext-auth-facebook": "^0.1.0",
"flarum/flarum-ext-auth-github": "^0.1.0",
"flarum/flarum-ext-auth-twitter": "^0.1.0",
"flarum/flarum-ext-bbcode": "^0.1.0",
"flarum/flarum-ext-emoji": "^0.1.0",
"flarum/flarum-ext-english": "^0.1.0",
"flarum/flarum-ext-flags": "^0.1.0",
"flarum/flarum-ext-likes": "^0.1.0",
"flarum/flarum-ext-lock": "^0.1.0",
"flarum/flarum-ext-markdown": "^0.1.0",
"flarum/flarum-ext-mentions": "^0.1.0",
"flarum/flarum-ext-pusher": "^0.1.0",
"flarum/flarum-ext-sticky": "^0.1.0",
"flarum/flarum-ext-subscriptions": "^0.1.0",
"flarum/flarum-ext-suspend": "^0.1.0",
"flarum/flarum-ext-tags": "^0.1.0",
"zendframework/zend-stratigility": "1.2.1",
"flagrow/byobu": "^0.1.0@beta",
"sijad/flarum-ext-spoiler-alert": "^0.1.0",
"clarkwinkelmann/flarum-ext-emojionearea": "^0.1.1",
"xengine/flarum-ext-markdown-editor": "^1.3",
"davis/flarum-ext-customfooter": "^0.1.0@beta",
"sijad/flarum-ext-pages": "^0.1.0@beta",
"avatar4eg/flarum-ext-users-list": "^0.1.1",
"s9e/flarum-ext-mediaembed": "^0.3.2",
"sijad/flarum-ext-links": "^0.1.0@beta",
"flagrow/flarum-ext-image-upload": "^0.3.1"
},
"require-dev": {
"franzl/studio": "^0.11.0"
},
"config": {
"preferred-install": "dist"
},
"minimum-stability": "beta",
"prefer-stable": true
}