This is the connection string I use for all my BDC files.  Once the file is imported into the BDC, I then assign permission accordingly.

<LobSystemInstances>
<LobSystemInstance Name=”BDC_Unique_Name_RI”>
<Properties>
<Property Name=”DatabaseAccessProvider” Type=”System.String”>SqlServer</Property>
<Property Name=”AuthenticationMode” Type=”System.String”>RevertToSelf</Property>
<Property Name=”RdbConnection Data Source” Type=”System.String”>Database_Server_Name</Property>
<Property Name=”RdbConnection Initial Catalog” Type=”System.String”>Database_Name</Property>
<Property Name=”RdbConnection Integrated Security” Type=”System.String”>SSPI</Property>
<Property Name=”RdbConnection Pooling” Type=”System.String”>false</Property>
<Property Name=”RdbConnection Trusted_Connection” Type=”System.String”>false</Property>
<Property Name=”RdbConnection User ID” Type=”System.String”>User_Name</Property>
<Property Name=”RdbConnection Password” Type=”System.String”>User_Password</Property>
</Properties>
</LobSystemInstance>
</LobSystemInstances>

In SQL Server I assing User_Name Select permissions on the items referenced in the BDC file.

NOTE

Make sure your default search account has access to the BDC file once it’s imported.  I applied Execute and Selectable in clients permissions on both the Application and Entity.

“How do I find the default accout?”

Shared Services Administration: SSP1 > Search Administration > Default content access account