AWS direct connect BGP to own or not

For AWS direct connect x EQUINIX, we are using private or public VIF?

If you want to access resources in the VPC via DirectConnect, use Private VIF.
Public VIF is used to access AWS services outside the VPC, such as S3 and DynamoDB.
It would be better to decide depending on what the customer does via DirectConnect.

Using the AWS direct connect, do we customer need to apply BGP ASN from local Telco? How to deal with BGP ASN anyway?

I don't think it is necessary to issue an AS number from the local phone company since the setup is done using a private AS number.
https://aws.amazon.com/directconnect/faqs/?nc1=h_ls

A public or private ASN. If you are using a public ASN, you must own it. If you are using a private ASN, it must be in the 64512 to 65535 range.