Или вы можете просто преобразовать строку расширений в String ^
vector <string> extensions = { "*.mp4", "*.avi", "*.flv" };
for (int i = 0; i < extensions.size(); ++i)
{
String^ ext = gcnew String(extensions[i].c_str());;
String^ path = "C:\\Users\\Eric\\Videos";
array<String^>^files = Directory::GetFiles(path,ext);
Console::WriteLine(ext);
cout << " " << (files->Length) << endl;
}